picos.expressions.exp_sumxtr¶

Implements SumExtremes.

Outline¶

Classes¶

Sum of the $k$ largest or smallest elements or eigenvalues.

Classes¶

SumExtremes¶

class picos.expressions.exp_sumxtr.SumExtremes(**kwargs)[source]¶

Bases: picos.expressions.expression.Expression

Sum of the $k$ largest or smallest elements or eigenvalues.

Definition

Let $k \in \mathbb{Z}_{\geq 1}$ .

If $x$ is an $n$ -dimensional real vector or matrix and eigenvalues == False, then this is the sum of the $k \leq n$ largest or smallest scalar elements of $x$ , depending on the truth value of largest.

Special cases:
- If $k = 1$ , this is either the largest element $\max_{i = 1}^n \operatorname{vec}(x)_i$ or the smallest element $\min_{i = 1}^n \operatorname{vec}(x)_i$ of $x$ .
- If $k = n$ , this is the sum of all elements $\langle x, 1 \rangle$ of $x$ .
If $X$ is an $n \times n$ hermitian matrix and eigenvalues == True, then this is the sum of the $k \leq n$ largest or smallest eigenvalues of $X$ , depending on the truth value of largest. Recall that the eigenvalues of a hermitian matrix are real.

Special cases:
- If $k = 1$ , this is either the largest eigenvalue $\lambda_{\max}(X)$ or the smallest eigenvalue $\lambda_{\min}(X)$ of $X$ .
- If $k = n$ , this equals the trace $\operatorname{tr}(X)$ .

If the given $k$ exceeds the $n$ of either case, then $k$ is silently clipped to $n$ .

__init__(x, k, largest, eigenvalues=False)[source]¶

Construct a SumExtremes.

Parameters

x (ComplexAffineExpression) – The affine expression to take a sum over.
k (int) – Number of summands.
largest (bool) – Whether to sum over the largest (eigen)values as opposed to the smallest.
eigenvalues (bool) – Whether to sum eigenvalues instead of elements.

property eigenvalues¶: Whether the sum concerns eigenvalues as opposed to elements.

property full¶: Whether the sum concerns all (eigen)values of the expression.

property k¶: Number of (eigen)values to sum.

property largest¶: Whether the sum concerns largest values as opposed to smallest.

property x¶: The expression under the sum.